\subsection{The DFT basis-set correction in a nutshell}
The basis-set correction investigated here proposes to use the RSDFT formalism to capture a part of the short-range correlation effects missing in a finite one-electron basis-set.
In a nutshell, this formalism relies on 1) the definition of a complementary density functional aiming at describing the correlation effects absent in a finite basis-set, 2) the definition of an \textit{effective non divergent interaction} as the real-space representation of the coulomb operator projected in a finite basis-set,
3) the fit of such an effective interaction with a long-range interaction through the definition of a \textit{range-separation parameter varying in space}, 4) the use of a correlation functional from RSDFT with a \textit{multi-determinant} reference evaluated with the range-separation parameter varying in space.
More details can be found in \cite{GinPraFerAssSavTou-JCP-18}.
\subsubsection{Definition of basis-set dependent complementary functional}
\caption{Dissociation energy ($D_e$) in kcal/mol of the C$_2$ molecule computed using FCIQMC, CIPSI, FCIQMC+F$_{12}$, CIPSI+LDA$_{\rm HF}$ and CIPSI+LDA$_{\text{HF-val}}$ (valence only interaction and density) in the Dunnng cc-pVXZ (VXZ) basis sets.}
